Optical mask for all-optical extended depth-of-field for imaging systems under incoherent illumination

ABSTRACT

A mask for enhancing the depth of focus of an optical imaging system is designed by optimizing an optical property (transmittance or reflectance) of the mask relative to the intensity distribution in the system&#39;s image plane. Preferably, a desired PSF intensity is selected, a desired misfocus parameter range is selected, and the optical property is adjusted to minimize a measure of the departure of the system&#39;s PSF intensity, as computed from the mask&#39;s optical property, from the desired PSF intensity, over the entire misfocus parameter range. Most preferably, the desired PSF intensity is selected as the inverse Fourier transform of a desired OTF. Preferably, the mask is fabricated as a DOE.

FIELD AND BACKGROUND OF THE INVENTION

The present invention relates to optical imaging and, more particularly,to a mask for insertion in the pupil of an optical imaging system thatprovides extended depth of field for imaging scenes located in theextended depth of field region and illuminated by incoherent lightwithout needing to process the image created by the imaging system thatincorporates the mask.

Imaging systems are known to require accurate focus alignment.Conventional imaging systems, for example cameras, are very sensitive tomisfocus. When the object and image planes are not in conjugatepositions, the resultant image is severely degraded. Nevertheless, thereare many applications (e.g. barcode reading, computer or machine vision,surveillance cameras, etc.) that require imaging of objects locatedanywhere within an extended depth of field (DOF) region, while allowingreduced contrast and resolution.

Conventional solutions for imaging systems with an extended DOF involvepupil stopping and apodization. See, for example, M. Mino and Y. Okano,“Improvement in the OTF of a defocused optical system through the use ofshaded apertures”, Applied Optics vol. 10 pp. 2219-2225 (1971); J. O.Castaneda et al., “arbitrary high focal depth with a quasioptimum realand positive transmittance apodizer”, Applied Optics vol. 28 pp.2666-2669 (1989); J. O. Castaneda and L. R. Berriel-Valdos, “Zone platefor arbitrary high focal depth”, Applied Optics vol. 29 pp. 994-997(1990). The main disadvantages with such solutions are reducedresolution and low light throughput.

Recently, a hybrid opto-electronic approach has been proposed, to solvesuch problems. See, for example, E. R. Dowski, Jr. and W. T. Cathey,“Extended depth-of-field through wave-front coding”, Applied Optics vol.34 (1995) pp. 1859-1866, and J. van der Gracht et al., “Broadbandbehavior of an optical-digital focus-invariant system”, Optics Lettersvol. 21 no. 13 (1996) pp. 919-921. Both of these references areincorporated by reference for all purposes as if fully set forth herein.In that approach, a non-absorptive phase mask is used to severelyaberrate, or encode, the wavefront of the light wave at the pupil. Theaberration distorts the obtained image (sometimes referred to as the“intermediate image”), often so much that the intermediate image isunidentifiable. Nevertheless, the intermediate image is insensitive tomisfocus for a wide range of DOF. Image acquisition is followed by adigital signal-processing (DSP) step to recover the final image from theintermediate image. Another approach, by W. Chi and N. George(“Computational imaging with the logarithmic asphere: theory”, Journalof the Optical Society of America vol. 20 pp. 2260-2273 (2003)) producesdistorted images that change with misfocus position. An iterative filteris used in this case to restore the image so as to finally provideimaging with a certain enhanced depth of field. The main advantages ofsuch approaches are that there is no reduction in light powercollection, and that theoretically, one can restore the image details upto the optical cutoff spatial frequency. Practically, resolution islimited by CCD pixel size, as well as by the presence of noise, whichmay even be amplified in the processing stage.

We have proposed a phase mask that consists of sixteen spatiallymultiplexed Fresnel lenses. See E. Ben-Eliezer et al., “All-opticalextended depth of field imaging system”, Journal of Optics A: Pure andApplied Optics vol. 5 (2003) pp. S164-S169, which reference isincorporated by reference for all purposes as if fully set forth herein.Although the corresponding all-optical imaging system has an extendedDOF, this imaging system is suboptimal for scenes illuminated byincoherent light.

There is thus a widely recognized need for, and it would be highlyadvantageous to have, a mechanism, for extending the DOF, such mechanismbeing optimized for incoherent illumination and not requiringpostprocessing.

SUMMARY OF THE INVENTION

The present invention provides such a mechanism, in the form of anoptical element or mask that is inserted in the optical path of animaging system or that is etched, coated, overlayed or molded onexisting optical elements such as lenses, prisms or windows that arealready present in the optical path of the imaging system.

According to the present invention, a mask for an optical imaging systemis designed by optimizing an optical property (transmittance orreflectance) of the mask relative to the incident intensity distribution(rather than e.g. the incident optical field distribution) on the imageplane of the optical imaging system. The mask then is fabricated inaccordance with the optimized optical property. The mask overcomesmisfocus degradation in the optical imaging system.

Preferably, the optimization is done by selecting a desired point spreadfunction (PSF) intensity |h_(D)|², selecting a desired range of amisfocus parameter ψ, and adjusting the mask's optical property tominimize a measure of a departure from |h_(D)|² of a system PSFintensity |h|² that is computed from the mask's optical property. Theminimization should hold for any position within the selected range ofthe misfocus parameter. Because the measure is based on PSF intensitiesrather than on the PSFs themselves, it is independent of the phases ofthe PSFs. Most preferably, the measure is a minimum mean square errormeasure, but the scope of the present invention includes any suitablemeasure.

More preferably, |h_(D)|² is selected by steps including selecting adesired optical transfer function (OTF). |h_(D)|² is the inverse Fouriertransform of the desired OTF. Most preferably, the desired OTF is suchthat there are no phase differences between its spatial frequencycomponents in a pre-selected spatial frequency band of interest.

Preferably, the optimizing is effected by steps including simulatedannealing.

The scope of the present invention also includes a mask made accordingto the method of the present invention. The mask may be one dimensionalor two dimensional. A “one-dimensional mask” is a two-dimensionalstructure that performs a certain operation along only one axis (ordirection). The two-dimensional structure of a one-dimensional maskconsists e.g. of bars, lines or stripes so that there are no changes inthe direction orthogonal to the direction along which the mask operates.A “two-dimensional mask” operates in all directions in a plane, and mayoperate one way in one direction and another way in the orthogonaldirection.

Preferably, in the case of a one-dimensional mask, the phase of the maskis antisymmetric, in order to accommodate a larger range of positionswithin the depth of field under consideration.

Preferably, the two-dimensional mask is separable, e.g., the product ofthe transmittances of two orthogonal one-dimensional masks or twoorthogonal one-dimensional masks in tandem. Alternatively, thetwo-dimensional mask is radial, i.e., not varying along the angularcoordinate of a polar coordinate system.

Preferably, the one- or two-dimensional mask is real, which means thatthe phase of the mask transmittance is either 0 or π. Fabrication ofsuch a mask is eased, inasmuch as it requires the implementation of onlytwo phases.

Preferably, the one- or two-dimensional mask is a phase-only mask. Doingwithout amplitude variations or absorbing portions on the masksimplifies fabrication and increases light throughput.

Preferably, the one- or two- dimensional mask is fabricated as adiffractive optic element, in which phase and amplitude variations areobtained via the phenomenon of diffraction. Preferred fabricationmethods include etching, injection molding, deposition and casting ofphase or amplitude absorbing overlays on pre-existing surfaces.

The scope of the present invention also includes an optical element thatin turn includes a mask of the present invention. The optical properties(e.g. transmissivity or reflectvity) of the optical element incorporatethe features provided by the mask. Examples of such optical elementsinclude lenses, filters, windows and prisms.

The scope of the present invention also includes an optical imagingsystem that includes a mask of the present invention. Examples of suchoptical systems include general-purpose lenses, computer vision systems,automatic vision systems, barcode readers, cameras, mobile phonecameras, PC-mounted cameras, and surveillance security imaging systems.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is herein described, by way of example only, withreference to the accompanying drawings, wherein:

FIG. 1 illustrates a desired OTF;

FIGS. 2A-2C are plots of OTFs of an imaging system that uses aone-dimensional mask of the present invention, at three values of ψ,

FIG. 3 shows the absolute value and the phase of the transmittancefunction of the one-dimensional mask whose OTFs are plotted in FIGS.2A-2C;

FIGS. 4A-4D are simulated images of a spoke target, imaged using a clearpupil vs. a separable mask of the present invention;

FIGS. 5A and 5B show the amplitude and phase distributions of a radialmask transmittance function of the present invention;

FIGS. 6A-6F are magnitude and phase cross sections of the OTF of animaging system that uses the mask of FIG. 5, at three values of ψ,

FIGS. 7A-7D are simulated images of a spoke target, imaged using theseparable mask of FIGS. 4B and 4D vs. using the radial mask of FIGS. 5Aand 5B;

FIGS. 8A-8I are MTF plots for three radial masks of the presentinvention at three values of ψ,

FIGS. 9A-9L are simulated images of a spoke target, imaged using a clearpupil vs. the three radial masks whose MTFs are shown in FIGS. 8A-8I, atthree values of ψ,

FIG. 10 is a schematic block diagram of an imaging device of the presentinvention.

These Figures are only illustrative. The performance of an opticalimaging system varies according to its design requirements orassumptions. These Figures, therefore, illustrate a comparison of theperformance of one particular optical imaging system used withincoherently illuminated objects and used with an unmodifiedconventional aperture, vs. the performance of the same system butincluding a mask of the present invention in its aperture.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is of a design method for a mask that can be usedto extend the DOF of an incoherent imaging system.

In the following examples, the mask of the present invention is atransmissive optical element that is characterized by an amplitude andphase transmittance distribution. It is to be understood that the scopeof the present invention also includes folded optical arrangements inwhich the mask is a reflective optical element used in a reflection modeand having a controllable amplitude and phase reflectivity. Theperformance of an imaging system that uses a reflection-mode mask isidentical to the performance of an imaging system that uses anequivalent transmission-mode mask.

Furthermore, although the mask of the present invention is describedherein as an independent element introduced in the optical path of animaging system, it is well-understood by those skilled in the art thatthe properties of such a mask can be implemented by suitably modifyingthe surface properties or the bulk properties of an existing opticalelement such as a lens, a prism, a filter or a glass plate such that thetransmission of the modified element is equal to the combinedtransmissions of the original optical element and the mask of thepresent invention. Moreover, although one way of implementing thepresent invention is described herein, it is to be understood that thescope of the present invention includes all such implementations thatare consistent with the appended claims.

The principles and operation of an optical mask according to the presentinvention may be better understood with reference to the drawings andthe accompanying description.

As noted above, incoherent imaging systems are linear in intensity, incontrast to coherent imaging systems, which are linear in the opticalfield distribution. Thus, the phase of the resulting coherent pointspread function is not important at all when considering incoherentlyilluminated systems. To our knowledge, the only method that wassuggested to deal directly with incoherent illumination sources is thatof Dowski, Jr. and Cathey as cited above. Nevertheless, their methodrequires post-processing steps in order to reveal the features of theacquired image. We now disclose applicable criteria, tailored fordesigning all-optical incoherent imaging systems, that do not requirepost-processing steps. Specifically, we describe design considerationsfor obtaining an optimal mask transmittance function that compensatesfor misfocus occurrence, in the sense of a minimum mean square error(MMSE) criterion, applied directly over the intensity distribution inthe image plane, thus fitting incoherent illumination scenes.

It is well known that misfocus aberration manifests itself by theappearance of a quadratic phase term at the imaging system pupil, namelyG(u,v;ψ)=exp[jψ(u ² +v ²)]  (1)where u and v are the normalized coordinates of the pupil plane. Themisfocus parameter, ψ, is the maximal phase difference at the pupil edgedue to misfocus aberration, and is provided by the following expression(J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York,1996), p. 148): $\begin{matrix}{\psi = {\frac{\pi\quad L^{2}}{4\lambda}\quad\left( {\frac{1}{d_{obj}} + \frac{1}{d_{img}} - \frac{1}{f}} \right)}} & (2)\end{matrix}$In equation (2), L is the pupil dimension, λ is the wavelength, f is thelens focal length, d_(obj) is the distance from the object to the lensand d_(img) is the distance from the image to the lens. When misfocusoccurs, the phase factor, given in equation (1), multiplies the pupil ofthe imaging system. As a result of that multiplication, the object isnot imaged in the desired plane, where the detector is located, but in adifferent plane. As a result, the detector acquires a degraded image. Itcan be shown that our prior art phase mask, that is designed to handlesuch misfocus imaging conditions, is optimal in the sense of the MMSEcriterion with respect to the optical field distribution. To satisfythat criterion one has to minimize the expression: $\begin{matrix}{E^{2} = {\int_{\Delta\psi}\quad{{\mathbb{d}\psi}{\int_{\Omega}{\int{{{{h\quad\left( {x,{y;\psi}} \right)} - {h_{D}\left( {x,y} \right)}}}^{2}\quad{\mathbb{d}x}\quad{\mathbb{d}y}}}}}}} & (3)\end{matrix}$In equation (3), x and y are the coordinates of the image plane, h isthe PSF provided by the imaging system containing the mask and h_(D) isthe desired PSF. However, if one designs an imaging system to beoperated with incoherent illumination, the expression in equation (3) isinadequate, because the error, as derived in that expression, depends onthe phase of the point spread function, which is irrelevant for imagingsystems that use incoherent illumination. Therefore, one should insteadminimize a different expression, namely: $\begin{matrix}{E^{2} = {\int_{\Delta\psi}\quad{{\mathbb{d}\psi}{\int_{\Omega}{\int{\left( {{{h\quad\left( {x,{y;\psi}} \right)}}^{2} - {{h_{D}\left( {x,y} \right)}}^{2}} \right)^{2}\quad{\mathbb{d}x}\quad{\mathbb{d}y}}}}}}} & (4)\end{matrix}$In other words, one must minimize the integrated squared differencebetween the PSF intensities rather than the integrated squareddifference between the PSFs themselves.

Arbitrary changes of the phase of the PSF distribution that is derivedfor coherent illumination do not affect the incoherently illuminatedimage. Thus, the error calculated by equation (3) changes when thishappens, whereas the error obtained by equation (4) does not. Therefore,in some exemplary embodiments of the present invention, the mask of thepresent invention is designed by minimizing equation (4). The mask thatminimizes equation (4) cannot be found analytically; but a simulatedannealing algorithm can be used in order to calculate it.

Specifically, the mask transmittance function is parametrized as a setof pixel amplitudes and phases associated with the pixels of the mask.Starting with an arbitrary distribution of amplitudes and phases andapplying the simulated annealing algorithm, the mask that minimizes theerror defined in equation (4) is obtained. Simulated annealing isdescribed, for example, in S. Kirkpatrick et al., “Optimization bysimulated annealing”, Science vol. 220 pp. 671-680 (1983).

In some exemplary embodiments of the present invention, the search islimited to one-dimensional masks. A one-dimensional mask can be used inan imaging system that operates uni-dimensionally, for example to imageone-dimensional structures such as bar codes. Alternatively, twoorthogonal one-dimensional masks can be used in tandem, or atwo-dimensional mask of the present invention can be fabricated as asingle element whose transmittance is the product of two orthogonalone-dimensional distributions, so that the mask or masks act(s) as aseparable two-dimensional function.

We have determined that compensation for misfocus occurring on bothsides of the nominal “in-focus” position, represented by both positivevalues of ψ and negative values of ψ, is achieved when the transmittanceof the mask has an amplitude distribution that is symmetric and a phasedistribution that is anti-symmetric.

The simulated annealing process allows the user to influence theoptimization process by utilizing weight functions, in order to enhanceor suppress specific spatial frequency regions. This is needed in orderto achieve higher resolution, and to eliminate image contrast reversalsthat are not acceptable for high quality imagery.

The classic OTF for imaging under incoherent illumination has atriangular shape, as provided by a clear pupil in the “in-focus”condition (see e.g. J. W. Goodman, Introduction to Fourier Optics(McGraw-Hill, New York, 1996), p. 149 or E. Ben-Eliezer et al.,“All-optical extended depth of field imaging system”, Journal of OpticsA: Pure and Applied Optics vol. 5 (2003), FIG. 3), cannot be maintainedwhen an extended DOF is desired. It is well known that the OTF shapedetermines the contrast of the obtained image. It is expected that inorder to maintain maximum resolution over the entire extended DOF, oneshould be ready to accept lower contrast values than those provided bythe ideal “in focus” OTF curve. Referring now to the drawings, FIG. 1shows an example of a typical OTF that represents the “desired OTFcurve”. This desired OTF curve was chosen to deliver images with acontrast of at least 10% for all normalized spatial frequencies up to50% of the theoretical maximum that is attainable with a full aperturein an in-focus condition, followed by a smooth decline to zero at theends of the normalized spatial frequency range. We also required thatthe phase differences between spatial frequency components of thedesired OTF in the band of spatial frequencies of interest shouldvanish, so that image distortions due to relative phase shifts betweendifferent spatial frequencies are eliminated and thus high-qualityimages are obtained in such an all-optical system with no need forpost-processing steps. High-quality images are generated over the entirerange of misfocus positions |ψ|≦ψ_(max), as long as there are no phasedifferences between the spatial frequency components in the OTF regionof interest. The corresponding PSF intensity distribution is the inverseFourier transform of the desired OTF. The present invention is based onshaping the OTF to be close to the desired OTF.

FIGS. 2A-2C are plots of the magnitude of the OTF curves, called MTFcurves, as well as the respective phase of the OTF curves for severalvalues of ψ. A sample design was carried out for an imaging system witha DOF that occupies the interval ψ∈[−15,15]. FIG. 2A shows the magnitudeand phase of the OTF for ψ=0. FIG. 2B shows the magnitude and phase ofthe OTF for ψ=12. FIG. 2C shows the magnitude and phase of the OTF forψ=14. Note that all phases are expressed herein in units of radians. Onenotes that the phase essentially vanishes in the region where thecontrast is significant. Therefore, good quality images can be acquiredby this imaging system without any post-processing step.

We carried out computations of the MTF obtained with our prior art maskthat is optimal for the coherent point spread function case vs. thoseobtained with the mask of the present invention. We found that the maskof the present invention provides an increase of about 30% in thehighest obtainable resolution, when a contrast of 7% is assumed asbaseline. The one-dimensional mask transmittance function itself of thepresent invention is shown in FIG. 3.

Computer simulations using a spoke target object imaged by an imagingsystem that incorporates either a separable mask or just a plain clearpupil are shown, for visual comparison, in FIGS. 4A-4D. The outputimages are provided for two cases: the first is taken in an “in-focus”position, and the second is obtained for an object located in a positioncorresponding to ψ=±12. Assuming a low f_(#) and a nominal in-focusmagnification of −1, we disregarded the slight deviations inmagnification due to different object locations. The images of FIGS. 4Aand 4C correspond to the clear pupil imaging system, while the images ofFIGS. 4B and 4D correspond to the separable mask of the presentinvention. FIGS. 4A and 4B are for the “in-focus” condition, i.e. ψ=0,and FIGS. 4C and 4D are for ψ=±12.

One can readily see the improvement in the extent of depth of field thatthe separable mask provides with respect to a same sized, clear apertureimaging system.

The separable mask of the present invention provides high resolutionprimarily in the x and y direction, as can be observed in the resultspresented in FIGS. 4B and 4D, which match the MTF curve behaviorpresented earlier in FIGS. 2A-2C. We readily notice that the separablemask of the present invention improves the performance of the imagingsystem. However, as often occurs in case of separable pupil functions,the contrast in directions other than x and y is reduced, due tomultiplications of low MTF values in these directions. It is seen inFIG. 4D that the spatial frequencies for the ±45° orientations, forexample, exhibit very low contrast, so that the image quality in thesedirections is reduced.

The mathematical features of the one dimensional optimal masktransmittance function, shown in FIG. 3, reveal that this is an almostperfect real function, with negligible imaginary components. As such,one may consider an embodiment in which this function is represented asa real and symmetric function.

We can generate a radial mask transmittance function by rotating theseparable mask transmittance function around the origin of a polarcoordinate system and averaging over all values of the polar angle. Theresulting mask transmittance function has an amplitude distribution asshown in FIG. 5A, where light colors represents high transmittance anddark colors represent low transmittance. The phase of this masktransmittance function is shown in FIG. 5B, where white stands forregions of π phase shifts, and black defines regions with no phaseshift.

Alternatively, instead of averaging a separable distribution over polarangles, a radial distribution is generated directly by performing asimulated annealing procedure on a radial 2D distribution function.

FIGS. 6A-6F show the magnitudes (FIGS. 6A-6C) and phases (FIGS. 6D-6F)of the corresponding OTF cross sections, under in-focus conditions,i.e., ψ=0 (FIGS. 6A and 6D), misfocus condition ψ=±12 (FIGS. 6B and 6E)and misfocus condition ψ=±14 (FIGS. 6C and 6F).

A comparison between the MTF curves of the radial mask, shown in FIGS.6A-6C, and the MTF curves along the x or y directions of the separablemask, shown in FIG. 2A-2C, reveals that the resolution is improved inall directions when using the radial mask, when compared to the highestresolution obtained along the x and y directions achieved with theseparable mask. However, the contrast achieved with the radial mask incase of severe misfocus is lower than the contrast obtained with theseparable mask in the x and y directions.

The reason that the radial mask transmittance function works so well isclosely related to the fact that the one-dimensional mask transmittancefunction, with the phase quantized, is almost perfectly symmetric andreal. Thus, when one sums the rotated separable mask transmittancefunction in all orientations, the resulting amplitude and phase areradial, that is: $\begin{matrix}{{\overset{\sim}{M}\quad(r)} = {\frac{1}{2\pi}{\int_{\alpha}^{\alpha + {2\pi}}{{M_{sep}\left( {r,\theta} \right)}\quad{\mathbb{d}\theta}}}}} & (5)\end{matrix}$In equation (5), M_(sep)(r,θ) is the separable pupil function, and M(r)is the obtained radial pupil mask. This radial mask acts as a multifocal lens, and therefore ameliorates the sensitivity of theconventional imaging system to misfocus conditions, when placed at thepupil plane.

If an OTF other than the one illustrated in FIG. 1 is chosen, adifferent transmittance function is obtained. If the resultingtransmittance function is sufficiently close to being symmetric andreal, then rotational averaging as in equation (5) is possible, therebyachieving a radial transmittance function. If this is not the case, thetransmittance function can be used in a separable mask configuration.

Equation (5) is just one representation of the composite mask generationof the present invention. Many other summation, averaging, and othermathematical operations are possible for manufacturing and generation ofother masks that could achieve similar purpose, perhaps with slightlydifferent emphasis. Moreover, the simulated annealing approach can beused with the additional constraint that the resulting mask should berotationally symmetrical, thus obtaining the desired radial maskdirectly.

Simulated annealing can also be used to design amplitude-only radialdistributions, phase-only radial distributions, or distributions subjectto any other constraints imposed by the designer, such as restrictionson the number of quantization levels (e.g. requiring only two phaselevels).

One way to fabricate a complete radial mask, having both amplitudevariations and phase variations, is as a diffractive optic element(DOE). In each one of the DOE pixels we uncover a circle, whose area isproportional to the amplitude of the computed radial mask transmittancefunction in order to imitate the desired amplitude response of the mask(FIG. 5A) at that particular pixel. The phase variations, that consistof only two levels, i.e. a phase shift of π and no phase shift at all,are readily obtained by a single step etching procedure (reactive ion orwet).

One method of DOE fabrication is as follows. A transparent glass plateis covered with a layer of chrome and with a layer of photoresist.First, we expose the photoresist only over the circles that belong topixels in the regions that need to be etched in glass (where a π phaseshift should be created). The chrome layer that covers the glass inthese circles is removed, and through these holes, the glass itself isetched down to a depth that provides a phase shift of π. At the end ofthis step the DOE regions, where a phase of π is required, are readywith their associated amplitudes, which are the areas of the holesthrough which the glass was etched.

At this stage, the glass plate is covered again by a photoresist layer,and we expose only the circles in the regions where no phase shift isrequired. Thereafter, the chrome is removed from these circles, clearingup the transmission region with unetched glass.

The last step is the removal of the photoresist from the whole plate.The original chrome metal layer that covers the glass is not removedfrom the regions between the holes during the process, and is notremoved at the end of the fabrication as well, as opposed to thefabrication procedure of a phase-only DOE.

We now compare the coherent point spread function (PSF) of the DOEradial mask, where amplitude values were defined by the circle area ineach pixel, and the PSF that would have been achieved if we couldcontrol the transmission of each pixel uniformly over its entire area.We refer to the latter mask as “the ideal mask”. We show that they arealmost identical. To see that, we derive the PSF for both cases. LetA_(mn) and φ_(mn) be the amplitude and phase of the pixel {m,n}. Thenthe “ideal mask” can be expressed by: $\begin{matrix}\begin{matrix}{{M\quad\left( {x,y} \right)} = {\sum\limits_{m,n}{A_{mn}{rect}\quad\left( \frac{x - {\left( {m + 0.5} \right)\Delta}}{\Delta} \right)}}} \\{{rect}\left( \frac{y - {\left( {n + 0.5} \right)\Delta}}{\Delta} \right)\exp\quad\left( {j\quad\varphi_{mn}} \right)}\end{matrix} & (6)\end{matrix}$where Δ is the pixel size, and rect(x) is equal to unity in the region|x|<0.5 and zero elsewhere.

The DOE embodiment of the radial mask of FIGS. 5A and 5B, on the otherhand, is represented by the expression: $\begin{matrix}\begin{matrix}{{M\quad\left( {x,y} \right)} = {\sum\limits_{m,n}{{circ}\quad\left( \frac{\sqrt{\left\lbrack {{x\left( {m - 0.5} \right)}\quad\Delta} \right\rbrack^{2} + \left\lbrack {y - {\left( {n - 0.5} \right)\quad\Delta}} \right\rbrack^{2}}}{r_{mn}} \right)}}} \\{\exp\quad\left( {j\quad\varphi_{mn}} \right)}\end{matrix} & (7)\end{matrix}$where circ(r) is unity within the circle defined by a radius of r=1 andzero otherwise.

The radius of each circle, r_(mn), is related to the amplitude A_(mn) bythe expression: $\begin{matrix}{r_{mn} = {r_{\max}\sqrt{\frac{A_{mn}}{A_{\max}}}}} & (8)\end{matrix}$where A_(max) is the maximal amplitude of the mask (usually 1), andr_(max) is the radius that we attach to the maximal amplitude. Usually,r_(max) is slightly smaller than Δ/2, so that the circles in adjacentpixels are always separated and do not overlap.

The resulting PSF, obtained by the ideal mask, given in equation (6),can be expressed as: $\begin{matrix}\begin{matrix}{{{PSF}_{des}\left( {x_{0},y_{0}} \right)} = {\sum\limits_{m,n}{A_{mn}\Delta^{2}\sin\quad c\quad\left( {\Delta\quad\frac{x_{0}}{\lambda\quad d_{img}}} \right)\sin\quad{c\left( {\Delta\quad\frac{y_{0}}{\lambda\quad d_{img}}} \right)} \times}}} \\{\exp\left\{ {j\quad\left\lbrack {\varphi_{mn} + {2\pi\quad\Delta\quad\begin{pmatrix}{{\left( {m - 0.5} \right)\frac{x_{0}}{\lambda\quad d_{img}}} +} \\{\left( {n - 0.5} \right)\frac{y_{0}}{\lambda\quad d_{img}}}\end{pmatrix}}} \right\rbrack} \right\}}\end{matrix} & (9)\end{matrix}$where x_(o) and y_(o) are the spatial coordinates of the plane in whichthe PSF is measured. Similarly, the resulting PSF, obtained by the DOEembodiment of the radial mask of FIGS. 5A and 5B is represented by:$\begin{matrix}\begin{matrix}{{{PSF}_{circ}\left( {x_{0},y_{0}} \right)} = {\sum\limits_{m,n}{\pi\quad r_{mn}^{2}\frac{J_{1}\left( {2\pi\quad r_{mn}{\sqrt{x_{0}^{2} + y_{0}^{2}}/\lambda}\quad d_{img}} \right)}{\pi\quad r_{mn}{\sqrt{x_{0}^{2} + y_{0}^{2}}/\lambda}\quad d_{img}} \times}}} \\{\exp\left\{ {j\quad\begin{bmatrix}{{\varphi_{mn} + {2\pi\quad\Delta}}\quad} \\\left( {{\left( {m - 0.5} \right)\frac{x_{0}}{\lambda\quad d_{img}}} + {\left( {n - 0.5} \right)\frac{y_{0}}{\lambda\quad d_{img}}}} \right)\end{bmatrix}} \right\}} \\{\square}\end{matrix} & (10)\end{matrix}$

Because most of the PSF energy is located in the vicinity of the origin,we are allowed to use the approximation: $\begin{matrix}{\frac{J_{1}\left( {2\pi\quad r_{mn}{\sqrt{x_{0}^{2} + y_{0}^{2}}/\lambda}\quad d_{img}} \right)}{\pi\quad r_{mn}{\sqrt{x_{0}^{2} + y_{0}^{2}}/\lambda}\quad d_{img}} \approx {\sin\quad{c\left( {\Delta\quad{x_{o}/\lambda}\quad d_{img}} \right)}\sin\quad{c\left( {\Delta\quad{y_{o}/\lambda}\quad d_{img}} \right)}} \approx 1} & (11)\end{matrix}$and because we took the areas of the circles to be proportional to theideal mask pixel amplitudes, as expressed in equation (8), the PSFresults of equations (9) and (10) have the same shape in the vicinity ofthe origin, where most of the energy is located.

The corresponding intensity impulse responses are obtained from thecoherent PSFs of both masks by calculating their absolute values squared(See Goodman, p. 134).

Simulation results of spoke target images are shown in FIGS. 7A-7D.FIGS. 7A and 7C show the resulting images when one uses the separablemask. FIGS. 7B and 7D provide the results obtained for the radial mask.FIGS. 7A and 7B show the images in focus, while in FIGS. 7C and 7Dresults for misfocus condition of ψ=±15 are provided. It is clearlyrealized that contrast, as well as resolution, are improved when we usethe radial mask (FIGS. 7B and 7D). Moreover, spatial frequencies at allpolar angles behave in the same fashion, as expected from the MTF curvesof FIGS. 6A-6C.

We compared the behavior of the complete radial mask that has bothamplitude and phase variations to the “phase mask only” which retainsonly the phase component of the complete radial mask. We also comparethe behavior of the complete radial mask to that of the amplitude-onlymask obtained from only the amplitude of the complete radial mask. Wecalculated the MTF curves for both cases, and found out that theamplitude-only mask can handle a DOF up to about |ψ|≦12, which issmaller than the DOF that the phase-only mask can handle, which is|ψ|≦14. The main advantage of the phase-only mask over the completeradial mask and the amplitude-only mask is higher light throughput.Nevertheless, for both phase-only and amplitude-only masks there are noimage contrast reversals within the DOF of |ψ|≦14 for normalized spatialfrequencies in the range of |v|≦1. Therefore, the performances of thephase-only mask and of the amplitude-only mask are limited by poorcontrast. FIGS. 8A-8I are plots of the MTF curves for all three masks.FIGS. 8A, 8D and 8G show the MTF curves of the optimal mask (withvariations in both amplitude and phase). FIGS. 8B, 8E and 8H show theMTF curves of the phase only mask, and FIGS. 8C, 8F and 8I show the MTFcurves of the amplitude only mask. FIGS. 8A-8C show the results forin-focus condition, and FIGS. 8G-8I show the results obtained formisfocus of ψ=±15. The misfocus condition of ψ=±12 is shown in FIGS.8D-8F.

Simulation results for all the radial masks mentioned above arepresented in FIGS. 9A-9L. FIGS. 9A, 9E and 9I show the images obtainedwith a conventional imaging system, i.e. with clear aperture. FIGS. 9B,9F and 9J present the images that the same imaging system provides, whenone places the radial mask at the pupil. FIGS. 9C, 9G and 9K show theimages acquired by the imaging system when the phase-only mask ismounted at its aperture, and in FIGS. 9D, 9H and 9L are shown the imagesprovided when the amplitude-only mask is used. FIGS. 9A-9D present thesimulation images acquired in focus. FIGS. 9E-9H show the images takenin misfocus condition of ψ=±12, and FIGS. 9I-9L present the imagesobtained for an extreme misfocus condition of ψ=±15.

We conclude that for the examined examples, the complete radial optimalmask provides the best combination between contrast and resolution, atthe expense of reducing light throughput by more than 80%. Thephase-only radial mask of this example, which retains only the phase ofthe optimal radial mask, provides high light throughput (the reductionof light throughput is only 8%), as well as an acceptable resolution andcontrast for a misfocus condition up to |ψ|=14.

The manufacture of the phase mask of the above example is very easy toaccomplish, as it needs only coarse details and requires only two-phaselevels, that is 0 and π, so that a single binary mask is required forits production.

Such a phase mask has many applications, e.g. for general-purposelenses, computer vision, automatic vision, barcode readers, surveillancecameras, mobile phone cameras, monitors, etc.

FIG. 10 is a schematic high-level block diagram of a generalized imagingdevice 10 of the present invention, for imaging an object 20. Imagingdevice 10 includes optics 12, that are represented symbolically in FIG.10 as a convex lens, for projecting an image of object 20 onto adetector 16. Detector 16 may be an electronic detector (e.g. CCD orCMOS), a conventional photographic film or plate, or any otherlight-sensitive material. A mask 14 of the present invention is placedin or near the pupil of imaging device 10. If detector 16 is anelectronic detector, then electronics 18 are provided for, e.g.,digitizing the output of detector 16, storing the digitized output,displaying the digitized output, etc. Object 20 may be located anywherein the DOF region, as indicated by arrows 20′ and 20″.

While the invention has been described with respect to a limited numberof embodiments, it will be appreciated that many variations,modifications and other applications of the invention may be made.

1. A method of making a mask for an optical imaging system, comprisingthe steps of: (a) optimizing an optical property of the mask relative toan intensity distribution incident on an image plane of the opticalimaging system; and (b) fabricating the mask in accordance with saidoptical property.
 2. The method of claim 1, wherein the mask overcomesmisfocus degradation in the optical imaging system.
 3. The method ofclaim 1, wherein said optical property is a transmittance of the mask.4. The method of claim 1, wherein said optical property is a reflectanceof the mask.
 5. The method of claim 1, wherein said optimizing iseffected by steps including: (i) selecting a desired point spreadfunction intensity; (ii) selecting a desired misfocus parameter range;and (iii) adjusting said optical property to minimize a measure of adeparture of a system point spread function intensity from said desiredpoint spread function intensity over substantially all of said misfocusparameter range, said system point spread function intensity beingcomputed from said optical property.
 6. The method of claim 5, whereinsaid measure is a minimum mean square error measure.
 7. The method ofclaim 5, wherein said desired point spread function intensity isselected by steps including selecting a desired optical transferfunction, said desired point spread function intensity then being aninverse Fourier transform of said desired optical transfer function. 8.The method of claim 7, wherein said desired optical transfer functionlacks phase differences between spatial frequency components thereof ina pre-selected band of spatial frequencies.
 9. The method of claim 1,wherein said optimizing is effected by steps including simulatedannealing.
 10. A mask made according to the method of claim
 1. 11. Themask of claim 10, wherein the mask is one-dimensional
 12. The mask ofclaim 1 1, wherein a phase of said mask is antisymmetric.
 13. The maskof claim 10, wherein the mask is two-dimensional.
 14. The mask of claim13, wherein the mask is a separable mask.
 15. The mask of claim 13,wherein the mask is a radial mask.
 16. The mask of claim 10, wherein themask is real.
 17. The mask of claim 10, wherein the mask is a phase-onlymask.
 18. The mask of claim 10, wherein the mask is fabricated as adiffractive optic element.
 19. The mask of claim 18, wherein saiddiffractive optical element is fabricated by a method selected from thegroup consisting of etching, injection molding, deposition and overlaycasting.
 20. An optical element comprising the mask of claim
 10. 21. Theoptical element of claim 20, selected from the group consisting oflenses, filters, windows and prisms.
 22. An optical imaging systemcomprising the mask of claim
 10. 23. The optical imaging system of claim22, selected from the group consisting of general-purpose lenses,computer vision systems, automatic vision systems, barcode readers,cameras and surveillance security imaging systems.
 24. The opticalimaging system of claim 23, wherein said cameras are selected from thegroup consisting of mobile phone cameras and PC-mounted cameras.